Intervals Between the Frets: How Middle Eastern Music Measures Pitch Distance

Western music theory names intervals by counting scale steps — a "third" spans three letter names, a "fifth" spans five. The actual pitch distance is then qualified as major, minor, perfect, augmented, or diminished. This system assumes twelve equally spaced semitones per octave and works well within that framework. It does not work for makam.

Turkish makam theory and the related Arabic maqam tradition use intervals that fall between the cracks of the twelve-tone grid. Rather than qualifying intervals by their relationship to a major scale, these traditions name each interval type directly and measure it in Holdrian commas — the 53 equal divisions of the octave that form the theoretical backbone of the Arel-Ezgi-Uzdilek (AEU) system. Understanding these named intervals is the first step toward hearing and performing makam music accurately.

The five functional intervals

Established The AEU system recognises a set of named interval categories, each defined by a specific number of Holdrian commas. Five of these intervals carry the structural weight of makam construction — they are the building blocks from which every cesni (tetrachord and pentachord) is assembled:

Established Two additional intervals appear in theoretical descriptions but rarely serve as independent steps in cesni construction. The koma (1 comma, approximately 22.6 cents) is the smallest unit of the 53-tone system — an interval too small for most listeners to perceive as a discrete pitch change, but significant for defining the exact positions of perdeler. The eksik bakiye (3 commas, approximately 67.9 cents) is occasionally cited in extended interval tables.

What the comma values mean

Established Each Holdrian comma equals 1/53 of an octave, or approximately 22.64 cents. The cent values in the table above are simply the comma count multiplied by this constant. A bakiye of 4 commas is 4 x 22.64 = 90.6 cents. A tanini of 9 commas is 9 x 22.64 = 203.8 cents. These values closely approximate the Pythagorean intervals from which they derive: the Pythagorean limma (256:243) measures approximately 90.2 cents, nearly identical to the 4-comma bakiye; the Pythagorean whole tone (9:8) measures approximately 203.9 cents, a fraction of a cent from the 9-comma tanini.

Established This close correspondence is not coincidence — it is the reason the 53-tone system was adopted. As described in the Holdrian commas article, 53 stacked Pythagorean fifths nearly close the circle of fifths, making 53-TET an exceptionally precise approximation of Pythagorean tuning. The interval names in the AEU system are, in effect, labels for specific positions on this 53-step grid.

The intervals that live "between the frets"

Established Of the five functional intervals, two have no close equivalent in Western twelve-tone music: the kucuk mucennep (5 commas, 113.2 cents) and the buyuk mucennep (8 commas, 181.1 cents). These are the intervals that give this site its name. They fall between the frets of any instrument built for Western tuning — too large to be a semitone, too small to be a whole tone, or somewhere in the gap between the two.

Established Western musicologists sometimes label these "neutral seconds" — intervals that are neither major nor minor but somewhere in between. In the 24-tone equal temperament used in Arabic notation, a neutral second sits at exactly 150 cents (three quarter tones), splitting the difference between a semitone and a whole tone. The Turkish 5-comma interval at 113.2 cents and the 8-comma interval at 181.1 cents are not symmetrically placed around that midpoint — the 5-comma step is closer to a semitone, the 8-comma step is closer to a whole tone — but both occupy the same general territory of pitch space that Western theory has no standard name for.

Practice The audible effect is distinctive. When you hear a Rast tetrachord — T K S, whole tone then buyuk mucennep then kucuk mucennep — the third degree (Segah, at 17 commas or approximately 385 cents above the tonic) sits very close to a just major third (the 5:4 ratio at 386 cents). But it does not sound like a Western major third, because the two steps leading up to it (8 commas, then 5 commas) distribute the pitch distance differently from anything in twelve-tone music. When you hear an Ussak tetrachord — S K T, the reverse pattern — the second degree arrives after just 5 commas (113 cents), creating a narrow, haunting opening step that is the signature sound of many makam melodies. These intervals give the music its emotional character, and they cannot be approximated on a piano.

How intervals build cesni

Established Intervals do not exist in isolation in makam theory. They combine to form cesni — the tetrachords (dortlu) and pentachords (besli) that are the true structural units of a makam. A tetrachord always spans a perfect fourth (22 commas, 498.1 cents). A pentachord always spans a perfect fifth (31 commas, 701.9 cents). Every makam is assembled from a tetrachord joined to a pentachord, or occasionally from other combinations, and the character of each cesni comes entirely from how its constituent intervals are ordered.

Established Six tetrachords form the core of the AEU system. Each arranges its intervals to sum to 22 commas:

Established Notice that the first three cesni — Cargah, Buselik, and Kurdi — use only bakiye (4) and tanini (9), which are the intervals closest to Western semitones and whole tones. These cesni sound the most familiar to Western ears. The moment a kucuk mucennep (5) or buyuk mucennep (8) appears, as in Rast and Ussak, the sound moves into distinctly non-Western territory. And when the artik ikili (12) appears, as in Hicaz, the wide augmented second creates the sound many Western listeners associate (sometimes reductively) with "Middle Eastern music."

The Arabic framework: ajnas and quarter tones

Attested The Arabic maqam tradition uses a different theoretical language to describe closely related melodic structures. Where Turkish theory speaks of cesni measured in Holdrian commas, Arabic theory speaks of ajnas (singular: jins) — tetrachordal and pentachordal units that serve the same structural function. At the 1932 Cairo Congress on Arab Music, a 24-tone equal temperament was adopted as the standard notation system for Arabic music, dividing the octave into quarter tones of 50 cents each.

Attested The principal ajnas families — Ajam, Bayati, Hijaz, Kurd, Nahawand, Rast, Sikah, Saba, and Nawa Athar — correspond broadly to the Turkish cesni. The Arabic Rast jins and the Turkish Rast dortlu describe essentially the same melodic shape; Hijaz maps to Hicaz; Kurd maps to Kurdi; Nahawand maps to Buselik; Ajam maps to Cargah.

Practice However, the measurement systems diverge. A Turkish theorist describes the Rast tetrachord as T + K + S = 9 + 8 + 5 commas. An Arabic theorist describes the Rast jins using quarter-tone notation, typically as whole tone + three-quarter tone + three-quarter tone. The two descriptions point at the same general melodic shape, but the 53-comma grid and the 24-quarter-tone grid carve the pitch space differently, and neither grid fully captures what performers actually do. As Habib Hassan Touma has emphasised, the 24-tone system is a notational convention, not a prescription for exact intonation — just as the AEU 53-tone system is a theoretical model, not a set of rigid pitch targets.

Theory versus practice: the limits of the grid

Practice No grid — 12-tone, 24-tone, or 53-tone — captures the full reality of performed makam or maqam music. Skilled performers adjust their intonation continuously according to the seyir (melodic path) of the makam, the register, the emotional trajectory of a phrase, and the conventions of their lineage. A perdeler position that is notionally 5 commas above the tonic might be performed slightly higher or lower depending on whether the melody is ascending or descending, whether the phrase is approaching a cadence, or which makam is being performed.

Attested This flexibility has led some scholars to question the adequacy of the AEU system. Ozan Yarman has argued that a 79-tone tuning model better represents the intervals Turkish musicians actually use in performance. Others have observed that historical Ottoman practice did not conform to any single equal temperament. The 53-comma framework remains the standard pedagogical tool — it is what conservatories teach, what published makam guides use, and what Meshk implements as its default tuning model — but it is a map, not the territory.

Hearing the intervals: practical ear training

Practice For a student coming from Western music, the most important ear-training task is learning to hear and produce the 5-comma and 8-comma intervals. These are the sounds that are genuinely new — the bakiye (4 commas) is close enough to a semitone, and the tanini (9 commas) close enough to a whole tone, that Western-trained ears can approximate them. But the kucuk mucennep and buyuk mucennep require building new perceptual categories.

Practice A useful starting exercise: learn to sing or play a Rast tetrachord (T + K + S = 9 + 8 + 5 commas) and compare it to a Western major tetrachord (whole tone + whole tone + semitone = 9 + 9 + 4 commas). The first two degrees sound similar — both begin with a whole tone. The difference emerges at the third degree: in Rast, the second step is 8 commas (181 cents) instead of the Western 9 commas (204 cents), and the third step is 5 commas (113 cents) instead of 4 commas (91 cents). The Rast third degree sits at about 385 cents — almost exactly a just major third (386 cents) — but the path to it feels different because the steps are distributed differently. Learning to feel that difference in the steps, not just hear the destination pitch, is where genuine makam comprehension begins.

Practice The next step is the Ussak tetrachord (S + K + T = 5 + 8 + 9 commas), which opens with the narrow 5-comma interval. This opening step is the hardest interval for Western-trained musicians to produce accurately — the tendency is either to sing a semitone (too narrow by about 13 cents) or to overshoot toward a whole tone. Practising with a reference drone and a tuner that displays cents is essential until the interval becomes instinctive. Meshk provides exactly this kind of guided practice with real-time pitch feedback.